When demand exceeds supply in a normal market, the price rises until it reaches a market-clearing level. But in this country, as in other developed nations, Americans do not primarily pay for care with their own money. They pay with time.
How long does it take you on the phone to make an appointment to see a doctor? How many days do you have to wait before she can see you? How long does it take to get to the doctor's office? Once there, how long do you have to wait before being seen? These are all non-price barriers to care, and there is substantial evidence that they are more important in deterring care than the fee the doctor charges, even for low-income patients.
When demand exceeds supply, doctors have a great deal of flexibility about who they see and when they see them. Not surprisingly, they tend to see those patients first who pay the highest fees.